Mathematics Consists of...

...two mutually unintelligible languages, one spoken by mathematicians...

In his book on Differential Geometry, Loring Tu includes a story about a remark by Physics Nobel winner C. N. Yang.

“Gauge fields are deeply related to some profoundly beautiful ideas of contemporary mathematics, ideas that are the driving forces of part of the mathematics of the last 40 years, . . . , the theory of fiber bundles.” Convinced that gauge fields are related to connections on fiber bundles, he tried to learn the fiber-bundle theory from several mathematical classics on the subject, but “learned nothing. The language of modern mathematics is too cold and abstract for a physicist”

Tu represents that his book is intended to be intelligible to physicists, and prerequisites are just his previous book "Introduction to Manifolds,"  a bit of point set topology, and, evidently, some abstract algebra. Seven chapters into the present book, https://www.amazon.com/Differential-Geometry-Connections-Characteristic-Mathematics/dp/3319550829/ref=sr_1_1?dchild=1&keywords=Loring+Tu&qid=1592937822&s=books&sr=1-1, he gets around to finally talking about an example, vector bundles.

The set X(M) of all C∞ vector fields on a manifold M has the structure of a real vector space, which is the same as a module over the field R of real numbers. Let F =C∞(M) again be the ring of C∞ functions on M. Since we can multiply a vector field by a C∞ function, the vector space X(M) is also a module over F. Thus the set X(M) has two module structures, over R and over F. In speaking of a linear map:

X(M) → X(M) one should be careful to specify whether it is R-linear or F-linear.

 

An F-linear map is of course R-linear, but the converse is not true.

 

Given an open subset U of a manifold M, one can think of U ×Rr as a family of vector spaces Rr parametrized by the points in U. A vector bundle, speaking, is a family of vector spaces that locally “looks” like U ×Rr.

Definition 7.1. A C∞ surjection π : E →M is a C∞ vector bundle of rank r if

(i) For every p ∈ M, the set Ep :=π−1(p) is a real vector space of dimension r;

(ii) every point p ∈ M has an open neighborhood U such that there is a fiberpreserving diffeomorphism

φU : π−1(U)→U ×Rr

Of course my brain is not in the  same class as that of C. N. Yang, but that still seems a bit cold and abstract to me.  And isn't it just a bit perverse to designate the relevant field by R and the ring by F?

I guess I used to know what rings and fields were, but modules were a mystery and I had to look them up.  They are still a mystery, BTW - maybe a kind of damaged vector space?

Still, my dim, age weakened brain has some conceptions of a fiber bundle, though no doubt I will get a bunch of stuff wrong, so the mathematically adept can correct me if they will.

Think of a closed curve.  Think of extending the curve by lines or line segments through each point, forming a ribbon.  If this ribbon is untwisted, so it has two sides, it is a trivial fiber bundle on the curve.  if we give it a 180 twist to make a one-sided mobius strip, it a non-trivial fiber bundle.

Physicists like to define fields in space or spacetime, like electromagnetic fields, velocity fields, and gravitational fields. These too are sorts of fiber bundles over space or spacetime - I think.  

Sometimes these can be non-trivial, I think, which is when things get interesting and Lie Algebras and all the intricate machinery beloved of math dudes gets involved - or so I guess.